This invention relates to methods and apparatus for designing combinatorial experiments. There is currently a tremendous amount of activity directed toward the discovery and optimization of materials such as superconductors, zeolites, magnetic materials, phosphors, catalysts, thermoelectric materials, high and low dielectric materials, polymers, pharmacological compounds, semiconducting solids, and the like. These new materials are typically useful because they possess desirable levels of one or more superior physical (or other) characteristics, such as, for example, electrical conductivity, color, bio-inertness, fabrication cost, or any other property. A variety of fields (such as pharmacology, chemistry, materials science) focus on the development of new materials with superior properties. Unfortunately, even though the chemistry of both small molecules and extended solids has been extensively explored, few general principles have emerged that allow one to predict with certainty the composition, structure, and reaction pathways for synthesis of such materials. New materials are typically discovered through experimentation, rather than designed from existing principles.
A common challenge is understanding how two materials actually differ from each other. Any two materials might be similar in one or many ways (e.g. composition) but different in many other ways. Thus, one material may possess characteristics that are “better” (for a particular purpose) than those of another material for any number of reasons. One goal of experimental science is determining how the characteristics that define a material's behavior—which can be referred to as the material's “properties”) vary in response to changes in a variety of conditions such as concentration or amount of different chemical components of the material, processing temperature, pressure, annealing time, molecular weight, exposure time to radiation, or the like. These conditions can be referred to as parameters (or factors) and can generally include any variable whose value can change in a continuous or discontinuous fashion. Experimental studies typically examine the variation of a given property (e.g., smell) with a measured parameter (e.g., molecular weight), often with the implicit assumption that all other parameters are held constant (i.e., their values are identical for the compared samples). In the ideal case, two materials only differ by one parameter, and variation in the measured property is construed to be caused by variation in this parameter.
Unfortunately, it is impossible to completely determine how two materials are “different”. While variation in a given parameter (e.g., chemical composition) might be fairly obvious (one sample has 20% more nitrogen than the other), variation in another parameter might remain hidden (one sample has a slightly preferred grain orientation, vs. another sample's random orientation). The challenge is determining which parameters have a significant effect on the property of interest. This challenge requires the examination of the effects of many different parameters on the desired properties. Variation in each of these parameters creates a parameter space: a hyperspace bounded by all the relevant parameters that describe a material. A single material is thus defined by its coordinates within this parameter space—the values for each of these parameters for the given material. The goal of materials development is finding the coordinates of the material with the best set of desired properties. The commonly used analogy “looking for a needle in a haystack” can loosely describe this process: the parameter space is the “haystack”, and the material(s) with the best set of properties is(are) the needle(s).
The process of deciding where in the parameter space to make and measure samples is called “sampling” or “populating” the parameter space. Traditionally, the discovery and development of various materials has predominantly been a trial and error process carried out by scientists who generate one experiment at a time—in other words, each axis in the parameter space is sampled serially. This process suffers from low success rates, long time lines, and high costs, particularly as the desired materials increase in complexity. Nevertheless, these methods have been successful for developing materials whose properties are governed by a relatively small number of parameters.
One set of techniques, which can be categorized under the general lable of “Design of Experiments” (“DOE”) has departed from this serial sampling model. Most such techniques are predicated on an a priori assumption of smoothness in the response surface (i.e., that the response surface can be approximated by a smooth function such as a linear or quadratic equation). As a result of this presumed smoothness, a limited number of levels for each factor (e.g., 2 or 3 levels), possibly combined with statistical analyses, is assumed to be sufficient to estimate the behavior of the response surface. Other types of DOE methods can generate relatively large (by typical DOE standards) numbers of experimental points. So-called “fill factorial” designs enable users to select larger number of levels, thereby creating a grid sampling of the desired parameter space. However, because they are not constrained by the “smoothness” assumption discussed above, these methods typically simplify the experiment design by restricting the number of factors to a modest number (e.g., fewer than 10). In either case, DOE techniques typically limit sampling to a constant precision (number of levels), and offer only relatively simple constraints on factors. Using these assumptions, typical DOE techniques can be applicable to systems including a moderate number (e.g., up to 10) factors, for which a relatively small number of experiments can yield the optimum values. To put these numbers in context, a typical DOE set of experiments might investigate 5 factors, each of which takes two levels, requiring 25=32 experiments, which may be distributed throughout the parameter space, rather than simply arrayed along one or more of the parameter axes.
However, many properties can be a function of a large number of different, often unknown parameters. Additionally, the combined effects of parameter variation (particularly in complex chemical and/or materials systems) can be much more complicated than the discrete effects of varying one or two parameters by themselves, resulting in a response surface that is extremely jagged, nonlinear, or similarly varying in unknown, unpredictable ways. For such systems, a very large parameter space must be precisely examined in order to identify the material with the best properties. As a result, the discovery of new materials often depends largely on the ability to synthesize and analyze large numbers of new compounds over a very broad parameter space. For example, one commentator has noted that to search the system of organic compounds of up to thirty atoms drawn from just five elements—C, O, N, S and H—would require preparing a library of roughly 1063 samples (an amount that, at just 1 mg each, is estimated to require a total mass of approximately 1060 grams—roughly the mass of 1027 suns). See W. F. Maier, “Combinatorial Chemistry- Challenge and Chance for the Development of New Catalysts and Materials,” Angew. Chem. Int. Ed., 1999, 38, p. 1216. When materials characteristics vary as a function of process conditions as well as composition, the search becomes correspondingly more complex. One approach to the preparation and analysis of such large numbers of compounds has been the application of combinatorial methods.
In general, combinatorics refers to the process of creating vast numbers of discrete, diverse samples, by varying a set of starting parameters in all possible combinations. Since its introduction into the bio- and pharmaceutical industries in the late 80's, it has dramatically sped up the drug discovery process and is now becoming a standard practice in those industries. See, e.g., Chem. Eng. News, Feb.12,1996. Only recently have combinatorial techniques been successfully applied to the preparation of materials outside of these fields. See, e.g., E. Danielson et al., SCIENCE 279, pp. 837-839; E. Danielson et al., NATURE 389, by pp. 944-948, 1997; G. Briceno et al., SCIENCE 270, pp. 273-275, 1995; X. D. Xiang et al., SCIENCE 268, 1738-1740,1995. By using various rapid deposition techniques, array-addressing strategies, and processing conditions, it is now possible to generate hundreds to thousands of diverse materials on a substrate of only a few square inches. These materials include, e.g., high Tc superconductors, magnetoresistors, and phosphors. Using these techniques, it is now possible to create large libraries of chemically diverse compounds or materials, including biomaterials, organics, inorganics, internmetallics, metal alloys, and ceramics, using a variety of sputtering, ablation, evaporation, and liquid dispensing systems as disclosed, for example, in U.S. Pat. No.5,959,297, 6,004,617 and 6,030,917, each of which is incorporated by reference herein.
However, while existing experiment design techniques may be suited for more limited uses, such as optimization of processes that are relatively well-understood, such techniques are ill-equipped to address the vast parameter spaces, irregular response surfaces and large libraries typical of these combinatorial techniques. Accordingly, there is a need for techniques for designing combinatorial experiments that address such concerns.